LMFDB - Embedded newform 688.2.bg.c.353.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [688,2,Mod(17,688)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(688, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([0, 0, 38]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("688.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 688 = 2^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	688.bg (of order $21$, degree $12$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.49370765906$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$3$ over $\Q(\zeta_{21})$
Twist minimal:	no (minimal twist has level 43)
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			353.3
Character	$\chi$	$=$	688.353
Dual form			688.2.bg.c.497.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.93359 + 0.442167i) q^{3} +(-0.492700 - 0.335917i) q^{5} +(-2.18154 + 3.77854i) q^{7} +(5.54371 + 1.71001i) q^{9} +O(q^{10})$	\(q+(2.93359 + 0.442167i) q^{3} +(-0.492700 - 0.335917i) q^{5} +(-2.18154 + 3.77854i) q^{7} +(5.54371 + 1.71001i) q^{9} +(0.452993 + 1.98469i) q^{11} +(-0.130264 + 1.73826i) q^{13} +(-1.29685 - 1.20330i) q^{15} +(1.21261 - 0.826746i) q^{17} +(2.07906 - 0.641305i) q^{19} +(-8.07048 + 10.1201i) q^{21} +(-1.99563 + 1.85167i) q^{23} +(-1.69679 - 4.32336i) q^{25} +(7.48807 + 3.60606i) q^{27} +(7.48963 - 1.12888i) q^{29} +(0.208229 - 0.530558i) q^{31} +(0.451329 + 6.02256i) q^{33} +(2.34412 - 1.12887i) q^{35} +(1.98083 + 3.43090i) q^{37} +(-1.15074 + 5.04173i) q^{39} +(1.32534 + 1.66193i) q^{41} +(-3.81986 - 5.32998i) q^{43} +(-2.15696 - 2.70475i) q^{45} +(2.08348 - 9.12834i) q^{47} +(-6.01823 - 10.4239i) q^{49} +(3.92287 - 1.88915i) q^{51} +(-0.441351 - 5.88942i) q^{53} +(0.443502 - 1.13002i) q^{55} +(6.38267 - 0.962032i) q^{57} +(4.60418 + 2.21726i) q^{59} +(-3.57526 - 9.10961i) q^{61} +(-18.5551 + 17.2167i) q^{63} +(0.648092 - 0.812681i) q^{65} +(0.506577 - 0.156258i) q^{67} +(-6.67310 + 4.54964i) q^{69} +(-3.91319 - 3.63091i) q^{71} +(-1.05058 + 14.0191i) q^{73} +(-3.06604 - 13.4332i) q^{75} +(-8.48745 - 2.61803i) q^{77} +(1.82633 - 3.16330i) q^{79} +(5.99228 + 4.08546i) q^{81} +(-10.2305 - 1.54199i) q^{83} -0.875173 q^{85} +22.4706 q^{87} +(5.97021 + 0.899865i) q^{89} +(-6.28390 - 4.28429i) q^{91} +(0.845453 - 1.46437i) q^{93} +(-1.23978 - 0.382420i) q^{95} +(-2.28198 - 9.99799i) q^{97} +(-0.882576 + 11.7772i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9}+O(q^{10}) $ 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9	$ 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9} + 4 q^{11} + 3 q^{15} - 10 q^{17} - 10 q^{19} - 21 q^{21} - 4 q^{23} - 2 q^{25} + 4 q^{27} + 9 q^{29} - 40 q^{31} - 11 q^{33} - 11 q^{35} - 19 q^{37} + q^{39} - 28 q^{41} + 8 q^{43} - 46 q^{45} + 30 q^{47} + 6 q^{49} - 57 q^{51} - 24 q^{53} - 14 q^{55} + 52 q^{57} + q^{59} - 14 q^{61} - 47 q^{63} + 38 q^{65} - 66 q^{67} - 7 q^{69} + 33 q^{71} + 29 q^{73} + 55 q^{75} - 27 q^{77} + 17 q^{79} + 38 q^{81} + 23 q^{83} - 56 q^{85} + 86 q^{87} - 19 q^{89} + 13 q^{91} - 30 q^{93} - q^{95} - 31 q^{97} + 31 q^{99}+O(q^{100}) $ 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9 + 4 * q^11 + 3 * q^15 - 10 * q^17 - 10 * q^19 - 21 * q^21 - 4 * q^23 - 2 * q^25 + 4 * q^27 + 9 * q^29 - 40 * q^31 - 11 * q^33 - 11 * q^35 - 19 * q^37 + q^39 - 28 * q^41 + 8 * q^43 - 46 * q^45 + 30 * q^47 + 6 * q^49 - 57 * q^51 - 24 * q^53 - 14 * q^55 + 52 * q^57 + q^59 - 14 * q^61 - 47 * q^63 + 38 * q^65 - 66 * q^67 - 7 * q^69 + 33 * q^71 + 29 * q^73 + 55 * q^75 - 27 * q^77 + 17 * q^79 + 38 * q^81 + 23 * q^83 - 56 * q^85 + 86 * q^87 - 19 * q^89 + 13 * q^91 - 30 * q^93 - q^95 - 31 * q^97 + 31 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/688\mathbb{Z}\right)^\times$.

$n$	$431$	$433$	$517$
$\chi(n)$	$1$	$e\left(\frac{1}{21}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	2.93359	+	0.442167i	1.69371	+	0.255285i	0.923802	−	0.382872i	$-0.125065\pi$
$3$	2.93359	+	0.442167i	1.69371	+	0.255285i	0.769906	+	0.638157i	$0.220303\pi$
$4$		0			0
$5$	−0.492700	−	0.335917i	−0.220342	−	0.150227i	0.448119	−	0.893974i	$-0.352094\pi$
$5$	−0.492700	−	0.335917i	−0.220342	−	0.150227i	−0.668461	+	0.743747i	$0.733047\pi$
$6$		0			0
$7$	−2.18154	+	3.77854i	−0.824544	+	1.42815i	0.0777227	+	0.996975i	$0.475235\pi$
$7$	−2.18154	+	3.77854i	−0.824544	+	1.42815i	−0.902267	+	0.431178i	$0.858098\pi$
$8$		0			0
$9$	5.54371	+	1.71001i	1.84790	+	0.570003i
$10$		0			0
$11$	0.452993	+	1.98469i	0.136582	+	0.598407i	0.996171	+	0.0874207i	$0.0278624\pi$
$11$	0.452993	+	1.98469i	0.136582	+	0.598407i	−0.859589	+	0.510986i	$0.829280\pi$
$12$		0			0
$13$	−0.130264	+	1.73826i	−0.0361289	+	0.482106i	0.949566	+	0.313567i	$0.101524\pi$
$13$	−0.130264	+	1.73826i	−0.0361289	+	0.482106i	−0.985695	+	0.168539i	$0.946095\pi$
$14$		0			0
$15$	−1.29685	−	1.20330i	−0.334844	−	0.310690i
$16$		0			0
$17$	1.21261	−	0.826746i	0.294102	−	0.200515i	−0.407275	−	0.913305i	$-0.633521\pi$
$17$	1.21261	−	0.826746i	0.294102	−	0.200515i	0.701378	+	0.712790i	$0.252569\pi$
$18$		0			0
$19$	2.07906	−	0.641305i	0.476969	−	0.147125i	−0.0469436	−	0.998898i	$-0.514948\pi$
$19$	2.07906	−	0.641305i	0.476969	−	0.147125i	0.523912	+	0.851772i	$0.324472\pi$
$20$		0			0
$21$	−8.07048	+	10.1201i	−1.76112	+	2.20838i
$22$		0			0
$23$	−1.99563	+	1.85167i	−0.416117	+	0.386100i	−0.860260	−	0.509856i	$-0.829699\pi$
$23$	−1.99563	+	1.85167i	−0.416117	+	0.386100i	0.444143	+	0.895956i	$0.353508\pi$
$24$		0			0
$25$	−1.69679	−	4.32336i	−0.339358	−	0.864671i
$26$		0			0
$27$	7.48807	+	3.60606i	1.44108	+	0.693987i
$28$		0			0
$29$	7.48963	−	1.12888i	1.39079	−	0.209628i	0.589434	−	0.807817i	$-0.299351\pi$
$29$	7.48963	−	1.12888i	1.39079	−	0.209628i	0.801355	+	0.598189i	$0.204113\pi$
$30$		0			0
$31$	0.208229	−	0.530558i	0.0373990	−	0.0952911i	−0.910954	−	0.412508i	$-0.864653\pi$
$31$	0.208229	−	0.530558i	0.0373990	−	0.0952911i	0.948353	+	0.317217i	$0.102748\pi$
$32$		0			0
$33$	0.451329	+	6.02256i	0.0785662	+	1.04839i
$34$		0			0
$35$	2.34412	−	1.12887i	0.396228	−	0.190813i
$36$		0			0
$37$	1.98083	+	3.43090i	0.325647	+	0.564037i	0.981643	−	0.190727i	$-0.0610846\pi$
$37$	1.98083	+	3.43090i	0.325647	+	0.564037i	−0.655996	+	0.754764i	$0.727751\pi$
$38$		0			0
$39$	−1.15074	+	5.04173i	−0.184266	+	0.807324i
$40$		0			0
$41$	1.32534	+	1.66193i	0.206984	+	0.259549i	0.874478	−	0.485066i	$-0.161204\pi$
$41$	1.32534	+	1.66193i	0.206984	+	0.259549i	−0.667494	+	0.744615i	$0.732633\pi$
$42$		0			0
$43$	−3.81986	−	5.32998i	−0.582523	−	0.812814i
$43$	−3.81986	−	5.32998i	−0.582523	−	0.812814i
$44$		0			0
$45$	−2.15696	−	2.70475i	−0.321541	−	0.403200i
$46$		0			0
$47$	2.08348	−	9.12834i	0.303907	−	1.33151i	−0.560266	−	0.828312i	$-0.689301\pi$
$47$	2.08348	−	9.12834i	0.303907	−	1.33151i	0.864174	−	0.503193i	$-0.167842\pi$
$48$		0			0
$49$	−6.01823	−	10.4239i	−0.859747	−	1.48913i
$50$		0			0
$51$	3.92287	−	1.88915i	0.549312	−	0.264535i
$52$		0			0
$53$	−0.441351	−	5.88942i	−0.0606242	−	0.808974i	−0.941797	−	0.336181i	$-0.890865\pi$
$53$	−0.441351	−	5.88942i	−0.0606242	−	0.808974i	0.881173	−	0.472794i	$-0.156754\pi$
$54$		0			0
$55$	0.443502	−	1.13002i	0.0598017	−	0.152372i
$56$		0			0
$57$	6.38267	−	0.962032i	0.845405	−	0.127424i
$58$		0			0
$59$	4.60418	+	2.21726i	0.599413	+	0.288662i	0.708881	−	0.705328i	$-0.249200\pi$
$59$	4.60418	+	2.21726i	0.599413	+	0.288662i	−0.109468	+	0.993990i	$0.534915\pi$
$60$		0			0
$61$	−3.57526	−	9.10961i	−0.457765	−	1.16637i	−0.954433	−	0.298426i	$-0.903539\pi$
$61$	−3.57526	−	9.10961i	−0.457765	−	1.16637i	0.496668	−	0.867941i	$-0.334557\pi$
$62$		0			0
$63$	−18.5551	+	17.2167i	−2.33773	+	2.16910i
$64$		0			0
$65$	0.648092	−	0.812681i	0.0803859	−	0.100801i
$66$		0			0
$67$	0.506577	−	0.156258i	0.0618882	−	0.0190900i	−0.263656	−	0.964617i	$-0.584929\pi$
$67$	0.506577	−	0.156258i	0.0618882	−	0.0190900i	0.325545	+	0.945527i	$0.394452\pi$
$68$		0			0
$69$	−6.67310	+	4.54964i	−0.803347	+	0.547712i
$70$		0			0
$71$	−3.91319	−	3.63091i	−0.464410	−	0.430910i	0.412941	−	0.910758i	$-0.364501\pi$
$71$	−3.91319	−	3.63091i	−0.464410	−	0.430910i	−0.877352	+	0.479848i	$0.840692\pi$
$72$		0			0
$73$	−1.05058	+	14.0191i	−0.122961	+	1.64081i	0.505401	+	0.862885i	$0.331345\pi$
$73$	−1.05058	+	14.0191i	−0.122961	+	1.64081i	−0.628362	+	0.777921i	$0.716274\pi$
$74$		0			0
$75$	−3.06604	−	13.4332i	−0.354036	−	1.55113i
$76$		0			0
$77$	−8.48745	−	2.61803i	−0.967234	−	0.298352i
$78$		0			0
$79$	1.82633	−	3.16330i	0.205478	−	0.355899i	−0.744807	−	0.667280i	$-0.767458\pi$
$79$	1.82633	−	3.16330i	0.205478	−	0.355899i	0.950285	+	0.311381i	$0.100792\pi$
$80$		0			0
$81$	5.99228	+	4.08546i	0.665809	+	0.453941i
$82$		0			0
$83$	−10.2305	−	1.54199i	−1.12294	−	0.169256i	−0.438784	−	0.898592i	$-0.644591\pi$
$83$	−10.2305	−	1.54199i	−1.12294	−	0.169256i	−0.684155	+	0.729337i	$0.739829\pi$
$84$		0			0
$85$	−0.875173			−0.0949258
$86$		0			0
$87$	22.4706			2.40910
$88$		0			0
$89$	5.97021	+	0.899865i	0.632841	+	0.0953855i	0.457626	−	0.889145i	$-0.348700\pi$
$89$	5.97021	+	0.899865i	0.632841	+	0.0953855i	0.175215	+	0.984530i	$0.443938\pi$
$90$		0			0
$91$	−6.28390	−	4.28429i	−0.658731	−	0.449115i
$92$		0			0
$93$	0.845453	−	1.46437i	0.0876694	−	0.151848i
$94$		0			0
$95$	−1.23978	−	0.382420i	−0.127198	−	0.0392355i
$96$		0			0
$97$	−2.28198	−	9.99799i	−0.231700	−	1.01514i	−0.948230	−	0.317586i	$-0.897128\pi$
$97$	−2.28198	−	9.99799i	−0.231700	−	1.01514i	0.716530	−	0.697556i	$-0.245729\pi$
$98$		0			0
$99$	−0.882576	+	11.7772i	−0.0887023	+	1.18365i
$100$		0			0
$101$	8.19564	+	7.60445i	0.815497	+	0.756671i	0.972989	−	0.230849i	$-0.0741505\pi$
$101$	8.19564	+	7.60445i	0.815497	+	0.756671i	−0.157492	+	0.987520i	$0.550341\pi$
$102$		0			0
$103$	0.115290	−	0.0786031i	0.0113598	−	0.00774499i	−0.557627	−	0.830092i	$-0.688288\pi$
$103$	0.115290	−	0.0786031i	0.0113598	−	0.00774499i	0.568987	+	0.822347i	$0.307336\pi$
$104$		0			0
$105$	7.37582	−	2.27514i	0.719807	−	0.222031i
$106$		0			0
$107$	−3.64662	+	4.57272i	−0.352533	+	0.442062i	−0.926203	−	0.377024i	$-0.876947\pi$
$107$	−3.64662	+	4.57272i	−0.352533	+	0.442062i	0.573671	+	0.819086i	$0.305519\pi$
$108$		0			0
$109$	−0.578145	+	0.536440i	−0.0553762	+	0.0513816i	−0.707375	−	0.706838i	$-0.750121\pi$
$109$	−0.578145	+	0.536440i	−0.0553762	+	0.0513816i	0.651999	+	0.758220i	$0.273931\pi$
$110$		0			0
$111$	4.29392	+	10.9407i	0.407560	+	1.03845i
$112$		0			0
$113$	5.42511	+	2.61260i	0.510352	+	0.245772i	0.671294	−	0.741191i	$-0.265739\pi$
$113$	5.42511	+	2.61260i	0.510352	+	0.245772i	−0.160942	+	0.986964i	$0.551453\pi$
$114$		0			0
$115$	1.60525	−	0.241953i	0.149691	−	0.0225622i
$116$		0			0
$117$	−3.69458	+	9.41364i	−0.341564	+	0.870292i
$118$		0			0
$119$	0.478526	+	6.38549i	0.0438664	+	0.585357i
$120$		0			0
$121$	6.17687	−	2.97462i	0.561533	−	0.270420i
$122$		0			0
$123$	3.15316	+	5.46143i	0.284311	+	0.492441i
$124$		0			0
$125$	−1.27974	+	5.60692i	−0.114464	+	0.501498i
$126$		0			0
$127$	2.35751	+	2.95623i	0.209195	+	0.262323i	0.875349	−	0.483492i	$-0.160632\pi$
$127$	2.35751	+	2.95623i	0.209195	+	0.262323i	−0.666153	+	0.745815i	$0.732060\pi$
$128$		0			0
$129$	−8.84914	−	17.3250i	−0.779124	−	1.52538i
$130$		0			0
$131$	10.8237	+	13.5725i	0.945669	+	1.18583i	0.982453	+	0.186508i	$0.0597170\pi$
$131$	10.8237	+	13.5725i	0.945669	+	1.18583i	−0.0367848	+	0.999323i	$0.511712\pi$
$132$		0			0
$133$	−2.11235	+	9.25483i	−0.183164	+	0.802496i
$134$		0			0
$135$	−2.47803	−	4.29207i	−0.213275	−	0.369403i
$136$		0			0
$137$	−2.80268	+	1.34970i	−0.239449	+	0.115312i	−0.549760	−	0.835322i	$-0.685281\pi$
$137$	−2.80268	+	1.34970i	−0.239449	+	0.115312i	0.310311	+	0.950635i	$0.399567\pi$
$138$		0			0
$139$	−0.390211	−	5.20700i	−0.0330973	−	0.441652i	−0.988995	−	0.147951i	$-0.952732\pi$
$139$	−0.390211	−	5.20700i	−0.0330973	−	0.441652i	0.955897	−	0.293701i	$-0.0948869\pi$
$140$		0			0
$141$	10.1483	−	25.8575i	0.854644	−	2.17760i
$142$		0			0
$143$	−3.50891	+	0.528883i	−0.293430	+	0.0442275i
$144$		0			0
$145$	−4.06935	−	1.95969i	−0.337941	−	0.162744i
$146$		0			0
$147$	−13.0459	−	33.2404i	−1.07601	−	2.74162i
$148$		0			0
$149$	−14.9269	+	13.8502i	−1.22286	+	1.13465i	−0.236229	+	0.971698i	$0.575911\pi$
$149$	−14.9269	+	13.8502i	−1.22286	+	1.13465i	−0.986634	+	0.162953i	$0.947898\pi$
$150$		0			0
$151$	12.4488	−	15.6103i	1.01307	−	1.27035i	0.0506675	−	0.998716i	$-0.483865\pi$
$151$	12.4488	−	15.6103i	1.01307	−	1.27035i	0.962401	−	0.271632i	$-0.0875634\pi$
$152$		0			0
$153$	8.13612	−	2.50966i	0.657766	−	0.202894i
$154$		0			0
$155$	−0.280818	+	0.191458i	−0.0225558	+	0.0153783i
$156$		0			0
$157$	−12.6185	−	11.7083i	−1.00707	−	0.934423i	−0.00921861	−	0.999958i	$-0.502934\pi$
$157$	−12.6185	−	11.7083i	−1.00707	−	0.934423i	−0.997850	+	0.0655341i	$0.979125\pi$
$158$		0			0
$159$	1.30937	−	17.4723i	0.103840	−	1.38564i
$160$		0			0
$161$	−2.64307	−	11.5800i	−0.208303	−	0.912636i
$162$		0			0
$163$	−20.8543	−	6.43270i	−1.63344	−	0.503848i	−0.663699	−	0.748000i	$-0.731014\pi$
$163$	−20.8543	−	6.43270i	−1.63344	−	0.503848i	−0.969737	+	0.244152i	$0.921491\pi$
$164$		0			0
$165$	1.80071	−	3.11892i	0.140185	−	0.242808i
$166$		0			0
$167$	−7.00644	−	4.77691i	−0.542175	−	0.369649i	0.261049	−	0.965325i	$-0.415931\pi$
$167$	−7.00644	−	4.77691i	−0.542175	−	0.369649i	−0.803224	+	0.595677i	$0.796884\pi$
$168$		0			0
$169$	9.85023	+	1.48468i	0.757710	+	0.114206i
$170$		0			0
$171$	12.6223			0.965254
$172$		0			0
$173$	−11.0837			−0.842677			−0.421338	−	0.906904i	$-0.638439\pi$
$173$	−11.0837			−0.842677			−0.421338	+	0.906904i	$0.638439\pi$
$174$		0			0
$175$	20.0376	+	3.02018i	1.51470	+	0.228304i
$176$		0			0
$177$	12.5264	+	8.54034i	0.941540	+	0.641931i
$178$		0			0
$179$	−7.70817	+	13.3509i	−0.576135	+	0.997896i	0.419782	+	0.907625i	$0.362107\pi$
$179$	−7.70817	+	13.3509i	−0.576135	+	0.997896i	−0.995917	+	0.0902708i	$0.971227\pi$
$180$		0			0
$181$	−3.06551	−	0.945583i	−0.227857	−	0.0702847i	0.178724	−	0.983899i	$-0.442803\pi$
$181$	−3.06551	−	0.945583i	−0.227857	−	0.0702847i	−0.406581	+	0.913615i	$0.633279\pi$
$182$		0			0
$183$	−6.46036	−	28.3047i	−0.477564	−	2.09234i
$184$		0			0
$185$	0.176543	−	2.35580i	0.0129797	−	0.173202i
$186$		0			0
$187$	2.19014	+	2.03215i	0.160159	+	0.148606i
$188$		0			0
$189$	−29.9612	+	20.4272i	−2.17935	+	1.48586i
$190$		0			0
$191$	5.09606	−	1.57193i	0.368738	−	0.113741i	−0.104848	−	0.994488i	$-0.533436\pi$
$191$	5.09606	−	1.57193i	0.368738	−	0.113741i	0.473586	+	0.880748i	$0.342959\pi$
$192$		0			0
$193$	−4.19096	+	5.25530i	−0.301672	+	0.378285i	−0.909444	−	0.415827i	$-0.863492\pi$
$193$	−4.19096	+	5.25530i	−0.301672	+	0.378285i	0.607772	+	0.794112i	$0.292064\pi$
$194$		0			0
$195$	2.26057	−	2.09751i	0.161883	−	0.150206i
$196$		0			0
$197$	−1.77600	−	4.52518i	−0.126535	−	0.322406i	0.853542	−	0.521024i	$-0.174450\pi$
$197$	−1.77600	−	4.52518i	−0.126535	−	0.322406i	−0.980077	+	0.198618i	$0.936355\pi$
$198$		0			0
$199$	−20.4216	−	9.83452i	−1.44765	−	0.697151i	−0.465463	−	0.885067i	$-0.654112\pi$
$199$	−20.4216	−	9.83452i	−1.44765	−	0.697151i	−0.982185	+	0.187916i	$0.939827\pi$
$200$		0			0
$201$	1.55518	−	0.234406i	0.109694	−	0.0165337i
$202$		0			0
$203$	−12.0734	+	30.7625i	−0.847387	+	2.15911i
$204$		0			0
$205$	−0.0947264	−	1.26404i	−0.00661598	−	0.0882841i
$206$		0			0
$207$	−14.2295	+	6.85259i	−0.989022	+	0.476288i
$208$		0			0
$209$	2.21459	+	3.83578i	0.153186	+	0.265327i
$210$		0			0
$211$	4.22277	−	18.5012i	0.290707	−	1.27367i	−0.592836	−	0.805323i	$-0.701992\pi$
$211$	4.22277	−	18.5012i	0.290707	−	1.27367i	0.883543	−	0.468349i	$-0.155151\pi$
$212$		0			0
$213$	−9.87422	−	12.3819i	−0.676571	−	0.848393i
$214$		0			0
$215$	0.0916113	+	3.90923i	0.00624784	+	0.266608i
$216$		0			0
$217$	1.55047	+	1.94423i	0.105253	+	0.131983i
$218$		0			0
$219$	−9.28074	+	40.6616i	−0.627134	+	2.74766i
$220$		0			0
$221$	1.27914	+	2.21553i	0.0860441	+	0.149033i
$222$		0			0
$223$	8.70607	−	4.19262i	0.583002	−	0.280759i	−0.119046	−	0.992889i	$-0.537984\pi$
$223$	8.70607	−	4.19262i	0.583002	−	0.280759i	0.702048	+	0.712130i	$0.252269\pi$
$224$		0			0
$225$	−2.01355	−	26.8690i	−0.134237	−	1.79126i
$226$		0			0
$227$	6.45065	−	16.4360i	0.428144	−	1.09089i	−0.540258	−	0.841500i	$-0.681673\pi$
$227$	6.45065	−	16.4360i	0.428144	−	1.09089i	0.968402	−	0.249394i	$-0.0802316\pi$
$228$		0			0
$229$	0.234167	−	0.0352950i	0.0154742	−	0.00233236i	−0.141301	−	0.989967i	$-0.545128\pi$
$229$	0.234167	−	0.0352950i	0.0154742	−	0.00233236i	0.156775	+	0.987634i	$0.449890\pi$
$230$		0			0
$231$	−23.7411	−	11.4331i	−1.56205	−	0.752242i
$232$		0			0
$233$	2.54575	+	6.48646i	0.166777	+	0.424942i	0.989690	−	0.143226i	$-0.0457476\pi$
$233$	2.54575	+	6.48646i	0.166777	+	0.424942i	−0.822913	+	0.568168i	$0.807652\pi$
$234$		0			0
$235$	−4.09290	+	3.79765i	−0.266991	+	0.247732i
$236$		0			0
$237$	6.75641	−	8.47228i	0.438876	−	0.550333i
$238$		0			0
$239$	28.2108	−	8.70187i	1.82480	−	0.562877i	0.824807	−	0.565414i	$-0.191284\pi$
$239$	28.2108	−	8.70187i	1.82480	−	0.562877i	0.999997	+	0.00253697i	$0.000807545\pi$
$240$		0			0
$241$	14.3617	−	9.79162i	0.925117	−	0.630734i	−0.00421050	−	0.999991i	$-0.501340\pi$
$241$	14.3617	−	9.79162i	0.925117	−	0.630734i	0.929327	+	0.369257i	$0.120388\pi$
$242$		0			0
$243$	−2.50506	−	2.32435i	−0.160699	−	0.149107i
$244$		0			0
$245$	−0.536378	+	7.15746i	−0.0342679	+	0.457274i
$246$		0			0
$247$	0.843926	+	3.69748i	0.0536977	+	0.235265i
$248$		0			0
$249$	−29.3301	−	9.04715i	−1.85872	−	0.573340i
$250$		0			0
$251$	−9.78186	+	16.9427i	−0.617426	+	1.06941i	0.372528	+	0.928021i	$0.378491\pi$
$251$	−9.78186	+	16.9427i	−0.617426	+	1.06941i	−0.989954	+	0.141392i	$0.954842\pi$
$252$		0			0
$253$	−4.57900	−	3.12191i	−0.287879	−	0.196273i
$254$		0			0
$255$	−2.56740	−	0.386973i	−0.160777	−	0.0242332i
$256$		0			0
$257$	21.9755			1.37080			0.685398	−	0.728168i	$-0.259628\pi$
$257$	21.9755			1.37080			0.685398	+	0.728168i	$0.259628\pi$
$258$		0			0
$259$	−17.2851			−1.07404
$260$		0			0
$261$	43.4507	+	6.54914i	2.68953	+	0.405382i
$262$		0			0
$263$	17.5044	+	11.9343i	1.07937	+	0.735900i	0.966230	−	0.257679i	$-0.0829578\pi$
$263$	17.5044	+	11.9343i	1.07937	+	0.735900i	0.113137	+	0.993579i	$0.463910\pi$
$264$		0			0
$265$	−1.76090	+	3.04997i	−0.108171	+	0.187358i
$266$		0			0
$267$	17.1163	+	5.27967i	1.04750	+	0.323110i
$268$		0			0
$269$	0.540806	+	2.36943i	0.0329735	+	0.144466i	0.988735	−	0.149676i	$-0.0478231\pi$
$269$	0.540806	+	2.36943i	0.0329735	+	0.144466i	−0.955762	+	0.294143i	$0.904966\pi$
$270$		0			0
$271$	−1.48929	+	19.8731i	−0.0904677	+	1.20721i	0.748541	+	0.663089i	$0.230755\pi$
$271$	−1.48929	+	19.8731i	−0.0904677	+	1.20721i	−0.839008	+	0.544118i	$0.816864\pi$
$272$		0			0
$273$	−16.5400	−	15.3469i	−1.00105	−	0.928835i
$274$		0			0
$275$	7.81189	−	5.32606i	0.471075	−	0.321173i
$276$		0			0
$277$	14.3542	−	4.42768i	0.862460	−	0.266034i	0.168196	−	0.985754i	$-0.446206\pi$
$277$	14.3542	−	4.42768i	0.862460	−	0.266034i	0.694265	+	0.719720i	$0.255730\pi$
$278$		0			0
$279$	2.06162	−	2.58519i	0.123426	−	0.154771i
$280$		0			0
$281$	−24.0279	+	22.2946i	−1.43338	+	1.32998i	−0.573793	+	0.819000i	$0.694529\pi$
$281$	−24.0279	+	22.2946i	−1.43338	+	1.32998i	−0.859590	+	0.510985i	$0.829281\pi$
$282$		0			0
$283$	−4.57856	−	11.6660i	−0.272167	−	0.693470i	−0.999970	−	0.00768433i	$-0.997554\pi$
$283$	−4.57856	−	11.6660i	−0.272167	−	0.693470i	0.727804	−	0.685786i	$-0.240541\pi$
$284$		0			0
$285$	−3.46790	−	1.67005i	−0.205421	−	0.0989254i
$286$		0			0
$287$	−9.17094	+	1.38230i	−0.541344	+	0.0815944i
$288$		0			0
$289$	−5.42387	+	13.8198i	−0.319051	+	0.812930i
$290$		0			0
$291$	−2.27359	−	30.3390i	−0.133280	−	1.77850i
$292$		0			0
$293$	6.61965	−	3.18786i	0.386724	−	0.186237i	−0.230411	−	0.973093i	$-0.574007\pi$
$293$	6.61965	−	3.18786i	0.386724	−	0.186237i	0.617136	+	0.786857i	$0.288293\pi$
$294$		0			0
$295$	−1.52366	−	2.63906i	−0.0887112	−	0.153652i
$296$		0			0
$297$	−3.76488	+	16.4950i	−0.218460	+	0.957138i
$298$		0			0
$299$	−2.95872	−	3.71012i	−0.171107	−	0.214562i
$300$		0			0
$301$	28.4727	−	2.80591i	1.64114	−	0.161730i
$302$		0			0
$303$	20.6802	+	25.9322i	1.18805	+	1.48976i
$304$		0			0
$305$	−1.29854	+	5.68929i	−0.0743544	+	0.325768i
$306$		0			0
$307$	5.68990	+	9.85520i	0.324740	+	0.562466i	0.981460	−	0.191669i	$-0.0613899\pi$
$307$	5.68990	+	9.85520i	0.324740	+	0.562466i	−0.656720	+	0.754135i	$0.728057\pi$
$308$		0			0
$309$	0.372968	−	0.179612i	0.0212174	−	0.0102178i
$310$		0			0
$311$	−1.85648	−	24.7730i	−0.105271	−	1.40475i	−0.760511	−	0.649325i	$-0.775052\pi$
$311$	−1.85648	−	24.7730i	−0.105271	−	1.40475i	0.655240	−	0.755421i	$-0.272567\pi$
$312$		0			0
$313$	6.18554	−	15.7605i	0.349627	−	0.890836i	−0.642578	−	0.766221i	$-0.722135\pi$
$313$	6.18554	−	15.7605i	0.349627	−	0.890836i	0.992205	−	0.124616i	$-0.0397698\pi$
$314$		0			0
$315$	14.9255	−	2.24965i	0.840955	−	0.126754i
$316$		0			0
$317$	−21.3116	−	10.2631i	−1.19698	−	0.576435i	−0.274166	−	0.961682i	$-0.588402\pi$
$317$	−21.3116	−	10.2631i	−1.19698	−	0.576435i	−0.922813	+	0.385247i	$0.874116\pi$
$318$		0			0
$319$	5.63322	+	14.3532i	0.315400	+	0.803626i
$320$		0			0
$321$	−12.7196	+	11.8021i	−0.709939	+	0.658727i
$322$		0			0
$323$	1.99090	−	2.49651i	0.110777	−	0.138909i
$324$		0			0
$325$	7.73614	−	2.38628i	0.429124	−	0.132367i
$326$		0			0
$327$	−1.93323	+	1.31806i	−0.106908	+	0.0728887i
$328$		0			0
$329$	29.9466	+	27.7864i	1.65101	+	1.53191i
$330$		0			0
$331$	−1.80682	+	24.1103i	−0.0993117	+	1.32522i	0.695570	+	0.718458i	$0.255152\pi$
$331$	−1.80682	+	24.1103i	−0.0993117	+	1.32522i	−0.794882	+	0.606764i	$0.792467\pi$
$332$		0			0
$333$	5.11429	+	22.4072i	0.280261	+	1.22791i
$334$		0			0
$335$	−0.302080	−	0.0931793i	−0.0165044	−	0.00509093i
$336$		0			0
$337$	7.76921	−	13.4567i	0.423216	−	0.733031i	−0.573036	−	0.819530i	$-0.694235\pi$
$337$	7.76921	−	13.4567i	0.423216	−	0.733031i	0.996252	+	0.0864989i	$0.0275679\pi$
$338$		0			0
$339$	14.7598	+	10.0631i	0.801645	+	0.546552i
$340$		0			0
$341$	1.14732	+	0.172931i	0.0621309	+	0.00936472i
$342$		0			0
$343$	21.9745			1.18651
$344$		0			0
$345$	4.81613			0.259292
$346$		0			0
$347$	−13.6421	−	2.05621i	−0.732346	−	0.110383i	−0.227726	−	0.973725i	$-0.573129\pi$
$347$	−13.6421	−	2.05621i	−0.732346	−	0.110383i	−0.504620	+	0.863342i	$0.668367\pi$
$348$		0			0
$349$	11.1033	+	7.57012i	0.594348	+	0.405219i	0.822796	−	0.568337i	$-0.192413\pi$
$349$	11.1033	+	7.57012i	0.594348	+	0.405219i	−0.228449	+	0.973556i	$0.573365\pi$
$350$		0			0
$351$	−7.24370	+	12.5465i	−0.386640	+	0.669680i
$352$		0			0
$353$	7.28752	+	2.24790i	0.387876	+	0.119644i	0.482559	−	0.875864i	$-0.339708\pi$
$353$	7.28752	+	2.24790i	0.387876	+	0.119644i	−0.0946829	+	0.995507i	$0.530184\pi$
$354$		0			0
$355$	0.708343	+	3.10346i	0.0375950	+	0.164714i
$356$		0			0
$357$	−1.41965	+	18.9440i	−0.0751361	+	1.00262i
$358$		0			0
$359$	12.3345	+	11.4447i	0.650989	+	0.604029i	0.934939	−	0.354810i	$-0.115454\pi$
$359$	12.3345	+	11.4447i	0.650989	+	0.604029i	−0.283950	+	0.958839i	$0.591645\pi$
$360$		0			0
$361$	−11.7873	+	8.03646i	−0.620385	+	0.422972i
$362$		0			0
$363$	19.4357	−	5.99511i	1.02011	−	0.314661i
$364$		0			0
$365$	5.22686	−	6.55427i	0.273586	−	0.343066i
$366$		0			0
$367$	0.0534689	−	0.0496119i	0.00279105	−	0.00258972i	−0.678776	−	0.734345i	$-0.737489\pi$
$367$	0.0534689	−	0.0496119i	0.00279105	−	0.00258972i	0.681567	+	0.731756i	$0.261299\pi$
$368$		0			0
$369$	4.50541	+	11.4796i	0.234542	+	0.597603i
$370$		0			0
$371$	23.2162	+	11.1803i	1.20533	+	0.580455i
$372$		0			0
$373$	−25.4691	+	3.83884i	−1.31874	+	0.198768i	−0.770438	−	0.637514i	$-0.779963\pi$
$373$	−25.4691	+	3.83884i	−1.31874	+	0.198768i	−0.548299	+	0.836282i	$0.684725\pi$
$374$		0			0
$375$	−6.23344	+	15.8825i	−0.321893	+	0.820171i
$376$		0			0
$377$	0.986652	+	13.1660i	0.0508152	+	0.678081i
$378$		0			0
$379$	−14.4579	+	6.96257i	−0.742654	+	0.357643i	−0.766647	−	0.642069i	$-0.778076\pi$
$379$	−14.4579	+	6.96257i	−0.742654	+	0.357643i	0.0239929	+	0.999712i	$0.492362\pi$
$380$		0			0
$381$	5.60882	+	9.71477i	0.287349	+	0.497703i
$382$		0			0
$383$	−5.36944	+	23.5250i	−0.274365	+	1.20207i	0.630437	+	0.776241i	$0.282876\pi$
$383$	−5.36944	+	23.5250i	−0.274365	+	1.20207i	−0.904802	+	0.425833i	$0.859981\pi$
$384$		0			0
$385$	3.30232	+	4.14098i	0.168302	+	0.211044i
$386$		0			0
$387$	−12.0619	−	36.0798i	−0.613139	−	1.83404i
$388$		0			0
$389$	10.2266	+	12.8237i	0.518507	+	0.650187i	0.970291	−	0.241940i	$-0.0777837\pi$
$389$	10.2266	+	12.8237i	0.518507	+	0.650187i	−0.451784	+	0.892127i	$0.649212\pi$
$390$		0			0
$391$	−0.889063	+	3.89524i	−0.0449619	+	0.196991i
$392$		0			0
$393$	25.7509	+	44.6019i	1.29896	+	2.24987i
$394$		0			0
$395$	−1.96244	+	0.945061i	−0.0987410	+	0.0475512i
$396$		0			0
$397$	−0.575635	−	7.68131i	−0.0288903	−	0.385514i	−0.992842	−	0.119436i	$-0.961891\pi$
$397$	−0.575635	−	7.68131i	−0.0288903	−	0.385514i	0.963952	−	0.266078i	$-0.0857278\pi$
$398$		0			0
$399$	−10.2890	+	26.2158i	−0.515092	+	1.31243i
$400$		0			0
$401$	−1.39718	+	0.210591i	−0.0697719	+	0.0105164i	−0.183835	−	0.982957i	$-0.558851\pi$
$401$	−1.39718	+	0.210591i	−0.0697719	+	0.0105164i	0.114064	+	0.993473i	$0.463613\pi$
$402$		0			0
$403$	0.895123	+	0.431068i	0.0445892	+	0.0214730i
$404$		0			0
$405$	−1.58002	−	4.02581i	−0.0785116	−	0.200044i
$406$		0			0
$407$	−5.91198	+	5.48551i	−0.293046	+	0.271907i
$408$		0			0
$409$	−4.44166	+	5.56967i	−0.219626	+	0.275402i	−0.879423	−	0.476042i	$-0.842071\pi$
$409$	−4.44166	+	5.56967i	−0.219626	+	0.275402i	0.659796	+	0.751444i	$0.270642\pi$
$410$		0			0
$411$	−8.81869	+	2.72021i	−0.434994	+	0.134178i
$412$		0			0
$413$	−18.4222	+	12.5600i	−0.906497	+	0.618039i
$414$		0			0
$415$	4.52256	+	4.19632i	0.222004	+	0.205989i
$416$		0			0
$417$	1.15765	−	15.4477i	0.0566903	−	0.756479i
$418$		0			0
$419$	3.05031	+	13.3643i	0.149018	+	0.652889i	0.993159	+	0.116769i	$0.0372536\pi$
$419$	3.05031	+	13.3643i	0.149018	+	0.652889i	−0.844142	+	0.536120i	$0.819889\pi$
$420$		0			0
$421$	−1.78387	−	0.550250i	−0.0869404	−	0.0268176i	0.250980	−	0.967992i	$-0.419247\pi$
$421$	−1.78387	−	0.550250i	−0.0869404	−	0.0268176i	−0.337920	+	0.941175i	$0.609723\pi$
$422$		0			0
$423$	27.1598	−	47.0421i	1.32055	−	2.28726i
$424$		0			0
$425$	−5.63187	−	3.83975i	−0.273186	−	0.186255i
$426$		0			0
$427$	42.2206	+	6.36373i	2.04320	+	0.307962i
$428$		0			0
$429$	−10.5276			−0.508275
$430$		0			0
$431$	23.8277			1.14774			0.573871	−	0.818946i	$-0.305441\pi$
$431$	23.8277			1.14774			0.573871	+	0.818946i	$0.305441\pi$
$432$		0			0
$433$	−39.6397	−	5.97472i	−1.90496	−	0.287127i	−0.912442	−	0.409206i	$-0.865806\pi$
$433$	−39.6397	−	5.97472i	−1.90496	−	0.287127i	−0.992520	+	0.122079i	$0.961044\pi$
$434$		0			0
$435$	−11.0713	−	7.54826i	−0.530827	−	0.361912i
$436$		0			0
$437$	−2.96154	+	5.12954i	−0.141670	+	0.245379i
$438$		0			0
$439$	−7.92185	−	2.44357i	−0.378089	−	0.116625i	0.0998848	−	0.994999i	$-0.468153\pi$
$439$	−7.92185	−	2.44357i	−0.378089	−	0.116625i	−0.477974	+	0.878374i	$0.658629\pi$
$440$		0			0
$441$	−15.5384	−	68.0781i	−0.739923	−	3.24182i
$442$		0			0
$443$	2.35373	−	31.4083i	0.111829	−	1.49225i	−0.605682	−	0.795706i	$-0.707100\pi$
$443$	2.35373	−	31.4083i	0.111829	−	1.49225i	0.717511	−	0.696547i	$-0.245281\pi$
$444$		0			0
$445$	−2.63924	−	2.44886i	−0.125112	−	0.116087i
$446$		0			0
$447$	−49.9136	+	34.0305i	−2.36083	+	1.60959i
$448$		0			0
$449$	14.7459	−	4.54851i	0.695903	−	0.214658i	0.0734316	−	0.997300i	$-0.476605\pi$
$449$	14.7459	−	4.54851i	0.695903	−	0.214658i	0.622471	+	0.782643i	$0.286129\pi$
$450$		0			0
$451$	−2.69804	+	3.38324i	−0.127046	+	0.159310i
$452$		0			0
$453$	43.4220	−	40.2897i	2.04014	−	1.89298i
$454$		0			0
$455$	1.65691	+	4.22173i	0.0776771	+	0.197918i
$456$		0			0
$457$	8.36585	+	4.02878i	0.391338	+	0.188458i	0.619197	−	0.785236i	$-0.287458\pi$
$457$	8.36585	+	4.02878i	0.391338	+	0.188458i	−0.227859	+	0.973694i	$0.573173\pi$
$458$		0			0
$459$	12.0614	−	1.81797i	0.562980	−	0.0848555i
$460$		0			0
$461$	−7.64092	+	19.4688i	−0.355873	+	0.906750i	0.635068	+	0.772456i	$0.280972\pi$
$461$	−7.64092	+	19.4688i	−0.355873	+	0.906750i	−0.990942	+	0.134294i	$0.957123\pi$
$462$		0			0
$463$	−0.163595	−	2.18303i	−0.00760292	−	0.101454i	0.992111	−	0.125360i	$-0.0400087\pi$
$463$	−0.163595	−	2.18303i	−0.00760292	−	0.101454i	−0.999714	+	0.0239064i	$0.992390\pi$
$464$		0			0
$465$	−0.908460	+	0.437491i	−0.0421288	+	0.0202882i
$466$		0			0
$467$	−0.115483	−	0.200022i	−0.00534391	−	0.00925593i	0.863341	−	0.504621i	$-0.168368\pi$
$467$	−0.115483	−	0.200022i	−0.00534391	−	0.00925593i	−0.868685	+	0.495365i	$0.835034\pi$
$468$		0			0
$469$	−0.514690	+	2.25500i	−0.0237662	+	0.104126i
$470$		0			0
$471$	−31.8406	−	39.9268i	−1.46714	−	1.83973i
$472$		0			0
$473$	8.84799	−	9.99567i	0.406831	−	0.459602i
$474$		0			0
$475$	−6.30032	−	7.90035i	−0.289079	−	0.362493i
$476$		0			0
$477$	7.62423	−	33.4040i	0.349090	−	1.52946i
$478$		0			0
$479$	2.50477	+	4.33839i	0.114446	+	0.198226i	0.917558	−	0.397602i	$-0.130157\pi$
$479$	2.50477	+	4.33839i	0.114446	+	0.198226i	−0.803112	+	0.595828i	$0.796824\pi$
$480$		0			0
$481$	−6.22183	+	2.99627i	−0.283691	+	0.136618i
$482$		0			0
$483$	−2.63336	−	35.1398i	−0.119822	−	1.59891i
$484$		0			0
$485$	−2.23417	+	5.69256i	−0.101448	+	0.258486i
$486$		0			0
$487$	−12.8944	+	1.94351i	−0.584299	+	0.0880689i	−0.434538	−	0.900653i	$-0.643088\pi$
$487$	−12.8944	+	1.94351i	−0.584299	+	0.0880689i	−0.149761	+	0.988722i	$0.547850\pi$
$488$		0			0
$489$	−58.3336	−	28.0920i	−2.63794	−	1.27036i
$490$		0			0
$491$	−9.25153	−	23.5725i	−0.417516	−	1.06381i	−0.972696	−	0.232083i	$-0.925446\pi$
$491$	−9.25153	−	23.5725i	−0.417516	−	1.06381i	0.555180	−	0.831730i	$-0.312649\pi$
$492$		0			0
$493$	8.14873	−	7.56092i	0.367000	−	0.340527i
$494$		0			0
$495$	4.39099	−	5.50613i	0.197360	−	0.247482i
$496$		0			0
$497$	22.2563	−	6.86516i	0.998332	−	0.307945i
$498$		0			0
$499$	−7.17570	+	4.89231i	−0.321228	+	0.219010i	−0.713188	−	0.700973i	$-0.752750\pi$
$499$	−7.17570	+	4.89231i	−0.321228	+	0.219010i	0.391960	+	0.919982i	$0.371797\pi$
$500$		0			0
$501$	−18.4418	−	17.1115i	−0.823920	−	0.764486i
$502$		0			0
$503$	−2.14991	+	28.6886i	−0.0958599	+	1.27916i	0.717153	+	0.696916i	$0.245445\pi$
$503$	−2.14991	+	28.6886i	−0.0958599	+	1.27916i	−0.813013	+	0.582246i	$0.802174\pi$
$504$		0			0
$505$	−1.48353	−	6.49976i	−0.0660162	−	0.289236i
$506$		0			0
$507$	28.2400	+	8.71090i	1.25418	+	0.386864i
$508$		0			0
$509$	−13.2695	+	22.9835i	−0.588162	+	1.01873i	0.406311	+	0.913735i	$0.366815\pi$
$509$	−13.2695	+	22.9835i	−0.588162	+	1.01873i	−0.994473	+	0.104992i	$0.966518\pi$
$510$		0			0
$511$	−50.6796	−	34.5528i	−2.24193	−	1.52852i
$512$		0			0
$513$	17.8807	+	2.69509i	0.789453	+	0.118991i
$514$		0			0
$515$	−0.0832072			−0.00366655
$516$		0			0
$517$	19.0607			0.838290
$518$		0			0
$519$	−32.5149	−	4.90084i	−1.42725	−	0.215123i
$520$		0			0
$521$	−29.4870	−	20.1039i	−1.29185	−	0.880769i	−0.294654	−	0.955604i	$-0.595205\pi$
$521$	−29.4870	−	20.1039i	−1.29185	−	0.880769i	−0.997196	+	0.0748353i	$0.976157\pi$
$522$		0			0
$523$	−19.4445	+	33.6789i	−0.850249	+	1.47267i	0.0307344	+	0.999528i	$0.490215\pi$
$523$	−19.4445	+	33.6789i	−0.850249	+	1.47267i	−0.880983	+	0.473147i	$0.843118\pi$
$524$		0			0
$525$	57.4466	+	17.7199i	2.50717	+	0.773361i
$526$		0			0
$527$	−0.186136	−	0.815515i	−0.00810821	−	0.0355244i
$528$		0			0
$529$	−1.16495	+	15.5452i	−0.0506501	+	0.675878i
$530$		0			0
$531$	21.7327	+	20.1650i	0.943120	+	0.875087i
$532$		0			0
$533$	−3.06150	+	2.08730i	−0.132608	+	0.0904109i
$534$		0			0
$535$	3.33275	−	1.02802i	0.144087	−	0.0444450i
$536$		0			0
$537$	−28.5159	+	35.7578i	−1.23055	+	1.54307i
$538$		0			0
$539$	17.9620	−	16.6663i	0.773676	−	0.717866i
$540$		0			0
$541$	6.99876	+	17.8325i	0.300900	+	0.766681i	0.998728	+	0.0504159i	$0.0160547\pi$
$541$	6.99876	+	17.8325i	0.300900	+	0.766681i	−0.697828	+	0.716265i	$0.745850\pi$
$542$		0			0
$543$	−8.57483	−	4.12942i	−0.367981	−	0.177210i
$544$		0			0
$545$	0.465051	−	0.0700951i	0.0199206	−	0.00300255i
$546$		0			0
$547$	0.882978	−	2.24979i	0.0377534	−	0.0961942i	−0.910754	−	0.412950i	$-0.864498\pi$
$547$	0.882978	−	2.24979i	0.0377534	−	0.0961942i	0.948507	+	0.316756i	$0.102594\pi$
$548$		0			0
$549$	−4.24269	−	56.6147i	−0.181074	−	2.41626i
$550$		0			0
$551$	14.8474	−	7.15014i	0.632521	−	0.304606i
$552$		0			0
$553$	7.96843	+	13.8017i	0.338852	+	0.586909i
$554$		0			0
$555$	1.55956	−	6.83289i	0.0661997	−	0.290040i
$556$		0			0
$557$	1.65271	+	2.07243i	0.0700274	+	0.0878116i	0.815611	−	0.578600i	$-0.196401\pi$
$557$	1.65271	+	2.07243i	0.0700274	+	0.0878116i	−0.745584	+	0.666412i	$0.767829\pi$
$558$		0			0
$559$	9.76247	−	5.94559i	0.412909	−	0.251472i
$560$		0			0
$561$	5.52642	+	6.92991i	0.233326	+	0.292581i
$562$		0			0
$563$	1.07396	−	4.70535i	0.0452622	−	0.198307i	−0.947242	−	0.320520i	$-0.896142\pi$
$563$	1.07396	−	4.70535i	0.0452622	−	0.198307i	0.992504	+	0.122214i	$0.0389993\pi$
$564$		0			0
$565$	−1.79534	−	3.10961i	−0.0755303	−	0.130822i
$566$		0			0
$567$	−28.5095	+	13.7294i	−1.19729	+	0.576582i
$568$		0			0
$569$	0.904422	+	12.0687i	0.0379153	+	0.505945i	0.983541	+	0.180687i	$0.0578321\pi$
$569$	0.904422	+	12.0687i	0.0379153	+	0.505945i	−0.945625	+	0.325258i	$0.894549\pi$
$570$		0			0
$571$	10.2678	−	26.1619i	0.429694	−	1.09484i	−0.538052	−	0.842912i	$-0.680840\pi$
$571$	10.2678	−	26.1619i	0.429694	−	1.09484i	0.967746	−	0.251930i	$-0.0810652\pi$
$572$		0			0
$573$	15.6448	−	2.35807i	0.653570	−	0.0985099i
$574$		0			0
$575$	11.3916	+	5.48591i	0.475063	+	0.228778i
$576$		0			0
$577$	−8.48151	−	21.6105i	−0.353090	−	0.899659i	−0.991517	−	0.129976i	$-0.958510\pi$
$577$	−8.48151	−	21.6105i	−0.353090	−	0.899659i	0.638427	−	0.769682i	$-0.279585\pi$
$578$		0			0
$579$	−14.6183	+	13.5638i	−0.607515	+	0.563691i
$580$		0			0
$581$	28.1446	−	35.2923i	1.16764	−	1.46417i
$582$		0			0
$583$	11.4887	−	3.54381i	0.475815	−	0.146770i
$584$		0			0
$585$	4.98252	−	3.39703i	0.206002	−	0.140450i
$586$		0			0
$587$	25.3059	+	23.4805i	1.04449	+	0.969143i	0.999554	−	0.0298472i	$-0.00950208\pi$
$587$	25.3059	+	23.4805i	1.04449	+	0.969143i	0.0449330	+	0.998990i	$0.485693\pi$
$588$		0			0
$589$	0.0926704	−	1.23660i	0.00381842	−	0.0509532i
$590$		0			0
$591$	−3.20917	−	14.0603i	−0.132008	−	0.578364i
$592$		0			0
$593$	25.9112	+	7.99254i	1.06404	+	0.328214i	0.776868	−	0.629663i	$-0.216807\pi$
$593$	25.9112	+	7.99254i	1.06404	+	0.328214i	0.287177	+	0.957878i	$0.407283\pi$
$594$		0			0
$595$	1.90922	−	3.30687i	0.0782705	−	0.135569i
$596$		0			0
$597$	−55.5600	−	37.8802i	−2.27392	−	1.55033i
$598$		0			0
$599$	7.61082	+	1.14715i	0.310970	+	0.0468711i	0.302673	−	0.953094i	$-0.402121\pi$
$599$	7.61082	+	1.14715i	0.310970	+	0.0468711i	0.00829658	+	0.999966i	$0.497359\pi$
$600$		0			0
$601$	−31.0119			−1.26500			−0.632500	−	0.774560i	$-0.717971\pi$
$601$	−31.0119			−1.26500			−0.632500	+	0.774560i	$0.717971\pi$
$602$		0			0
$603$	3.07552			0.125245
$604$		0			0
$605$	−4.04256	−	0.609319i	−0.164354	−	0.0247723i
$606$		0			0
$607$	−3.94082	−	2.68680i	−0.159953	−	0.109054i	0.480698	−	0.876886i	$-0.340383\pi$
$607$	−3.94082	−	2.68680i	−0.159953	−	0.109054i	−0.640651	+	0.767832i	$0.721336\pi$
$608$		0			0
$609$	−49.0206	+	84.9061i	−1.98641	+	3.44057i
$610$		0			0
$611$	15.5960	+	4.81073i	0.630947	+	0.194621i
$612$		0			0
$613$	7.62154	+	33.3922i	0.307831	+	1.34870i	0.858003	+	0.513645i	$0.171705\pi$
$613$	7.62154	+	33.3922i	0.307831	+	1.34870i	−0.550171	+	0.835052i	$0.685438\pi$
$614$		0			0
$615$	0.281027	−	3.75005i	0.0113321	−	0.151216i
$616$		0			0
$617$	−16.7395	−	15.5320i	−0.673908	−	0.625295i	0.267104	−	0.963668i	$-0.413933\pi$
$617$	−16.7395	−	15.5320i	−0.673908	−	0.625295i	−0.941012	+	0.338372i	$0.890124\pi$
$618$		0			0
$619$	−13.4359	+	9.16045i	−0.540035	+	0.368190i	−0.802407	−	0.596777i	$-0.796448\pi$
$619$	−13.4359	+	9.16045i	−0.540035	+	0.368190i	0.262372	+	0.964967i	$0.415495\pi$
$620$		0			0
$621$	−21.6206	+	6.66909i	−0.867607	+	0.267621i
$622$		0			0
$623$	−16.4244	+	20.5956i	−0.658031	+	0.825145i
$624$		0			0
$625$	−14.5090	+	13.4624i	−0.580359	+	0.538494i
$626$		0			0
$627$	4.80064	+	12.2318i	0.191719	+	0.488492i
$628$		0			0
$629$	5.23847	+	2.52272i	0.208872	+	0.100587i
$630$		0			0
$631$	−1.89851	+	0.286155i	−0.0755786	+	0.0113916i	−0.186723	−	0.982413i	$-0.559787\pi$
$631$	−1.89851	+	0.286155i	−0.0755786	+	0.0113916i	0.111144	+	0.993804i	$0.464548\pi$
$632$		0			0
$633$	20.5685	−	52.4076i	0.817523	−	2.08302i
$634$		0			0
$635$	−0.168499	−	2.24846i	−0.00668667	−	0.0892274i
$636$		0			0
$637$	18.9034	−	9.10337i	0.748978	−	0.360689i
$638$		0			0
$639$	−15.4847	−	26.8203i	−0.612566	−	1.06099i
$640$		0			0
$641$	6.59005	−	28.8729i	0.260291	−	1.14041i	−0.660645	−	0.750698i	$-0.729717\pi$
$641$	6.59005	−	28.8729i	0.260291	−	1.14041i	0.920937	−	0.389712i	$-0.127426\pi$
$642$		0			0
$643$	6.44594	+	8.08296i	0.254203	+	0.318761i	0.892515	−	0.451017i	$-0.148939\pi$
$643$	6.44594	+	8.08296i	0.254203	+	0.318761i	−0.638312	+	0.769778i	$0.720367\pi$
$644$		0			0
$645$	−1.45979	+	11.5086i	−0.0574790	+	0.453150i
$646$		0			0
$647$	−9.51303	−	11.9290i	−0.373996	−	0.468976i	0.558841	−	0.829275i	$-0.311246\pi$
$647$	−9.51303	−	11.9290i	−0.373996	−	0.468976i	−0.932837	+	0.360299i	$0.882675\pi$
$648$		0			0
$649$	−2.31491	+	10.1423i	−0.0908681	+	0.398119i
$650$		0			0
$651$	3.68878	+	6.38915i	0.144575	+	0.250411i
$652$		0			0
$653$	1.77106	−	0.852898i	0.0693070	−	0.0333765i	−0.398909	−	0.916990i	$-0.630611\pi$
$653$	1.77106	−	0.852898i	0.0693070	−	0.0333765i	0.468216	+	0.883614i	$0.344897\pi$
$654$		0			0
$655$	−0.773602	−	10.3230i	−0.0302271	−	0.403353i
$656$		0			0
$657$	−29.7968	+	75.9210i	−1.16248	+	2.96196i
$658$		0			0
$659$	−29.3755	+	4.42764i	−1.14431	+	0.172476i	−0.693712	−	0.720252i	$-0.744026\pi$
$659$	−29.3755	+	4.42764i	−1.14431	+	0.172476i	−0.450594	+	0.892729i	$0.648788\pi$
$660$		0			0
$661$	10.6317	+	5.11995i	0.413525	+	0.199143i	0.629066	−	0.777352i	$-0.283437\pi$
$661$	10.6317	+	5.11995i	0.413525	+	0.199143i	−0.215542	+	0.976495i	$0.569152\pi$
$662$		0			0
$663$	2.77283	+	7.06505i	0.107688	+	0.274384i
$664$		0			0
$665$	4.14961	−	3.85028i	0.160915	−	0.149307i
$666$		0			0
$667$	−12.8562	+	16.1212i	−0.497794	+	0.624214i
$668$		0			0
$669$	27.3939	−	8.44989i	1.05911	−	0.326692i
$670$		0			0
$671$	16.4602	−	11.2224i	0.635439	−	0.433235i
$672$		0			0
$673$	−14.9166	−	13.8406i	−0.574993	−	0.533516i	0.338183	−	0.941080i	$-0.390188\pi$
$673$	−14.9166	−	13.8406i	−0.574993	−	0.533516i	−0.913176	+	0.407565i	$0.866378\pi$
$674$		0			0
$675$	2.88460	−	38.4923i	0.111028	−	1.48157i
$676$		0			0
$677$	5.30533	+	23.2442i	0.203900	+	0.893346i	0.968534	+	0.248881i	$0.0800628\pi$
$677$	5.30533	+	23.2442i	0.203900	+	0.893346i	−0.764634	+	0.644465i	$0.777080\pi$
$678$		0			0
$679$	42.7560	+	13.1885i	1.64082	+	0.506127i
$680$		0			0
$681$	26.1910	−	45.3641i	1.00364	−	1.73836i
$682$		0			0
$683$	−29.2625	−	19.9508i	−1.11970	−	0.763398i	−0.145560	−	0.989349i	$-0.546498\pi$
$683$	−29.2625	−	19.9508i	−1.11970	−	0.763398i	−0.974138	+	0.225952i	$0.927451\pi$
$684$		0			0
$685$	1.83426	+	0.276471i	0.0700836	+	0.0105634i
$686$		0			0
$687$	0.702556			0.0268042
$688$		0			0
$689$	10.2948			0.392202
$690$		0			0
$691$	15.4237	+	2.32475i	0.586746	+	0.0884377i	0.435704	−	0.900090i	$-0.356499\pi$
$691$	15.4237	+	2.32475i	0.586746	+	0.0884377i	0.151041	+	0.988527i	$0.451737\pi$
$692$		0			0
$693$	−42.5751	−	29.0272i	−1.61729	−	1.10265i
$694$		0			0
$695$	−1.55686	+	2.69657i	−0.0590552	+	0.102287i
$696$		0			0
$697$	2.98112	+	0.919554i	0.112918	+	0.0348306i
$698$		0			0
$699$	4.60007	+	20.1542i	0.173991	+	0.762304i
$700$		0			0
$701$	0.820522	−	10.9491i	0.0309907	−	0.413542i	−0.960036	−	0.279877i	$-0.909706\pi$
$701$	0.820522	−	10.9491i	0.0309907	−	0.413542i	0.991027	−	0.133665i	$-0.0426746\pi$
$702$		0			0
$703$	6.31852	+	5.86273i	0.238308	+	0.221117i
$704$		0			0
$705$	−13.6861	+	9.33100i	−0.515447	+	0.351426i
$706$		0			0
$707$	−46.6128	+	14.3781i	−1.75305	+	0.540746i
$708$		0			0
$709$	20.7337	−	25.9992i	0.778670	−	0.976421i	−0.221329	−	0.975199i	$-0.571040\pi$
$709$	20.7337	−	25.9992i	0.778670	−	0.976421i	0.999999	−	0.00122191i	$-0.000388946\pi$
$710$		0			0
$711$	15.5339	−	14.4134i	0.582568	−	0.540544i
$712$		0			0
$713$	0.566873	+	1.44437i	0.0212295	+	0.0540920i
$714$		0			0
$715$	1.90650	+	0.918122i	0.0712991	+	0.0343358i
$716$		0			0
$717$	86.6065	−	13.0538i	3.23438	−	0.487504i
$718$		0			0
$719$	8.42223	−	21.4595i	0.314096	−	0.800304i	−0.683431	−	0.730015i	$-0.739513\pi$
$719$	8.42223	−	21.4595i	0.314096	−	0.800304i	0.997527	−	0.0702882i	$-0.0223919\pi$
$720$		0			0
$721$	0.0454960	+	0.607102i	0.00169436	+	0.0226096i
$722$		0			0
$723$	46.4608	−	22.3743i	1.72789	−	0.832110i
$724$		0			0
$725$	−17.5889	−	30.4649i	−0.653235	−	1.13144i
$726$		0			0
$727$	6.21668	−	27.2370i	0.230564	−	1.01017i	−0.718610	−	0.695413i	$-0.755221\pi$
$727$	6.21668	−	27.2370i	0.230564	−	1.01017i	0.949174	−	0.314753i	$-0.101922\pi$
$728$		0			0
$729$	−19.8866	−	24.9370i	−0.736540	−	0.923592i
$730$		0			0
$731$	−9.03855	−	3.30516i	−0.334303	−	0.122246i
$732$		0			0
$733$	4.22218	+	5.29445i	0.155950	+	0.195555i	0.853668	−	0.520817i	$-0.174373\pi$
$733$	4.22218	+	5.29445i	0.155950	+	0.195555i	−0.697718	+	0.716372i	$0.745801\pi$
$734$		0			0
$735$	−4.73831	+	20.7599i	−0.174775	+	0.765740i
$736$		0			0
$737$	0.539600	+	0.934614i	0.0198764	+	0.0344269i
$738$		0			0
$739$	13.6045	−	6.55159i	0.500450	−	0.241004i	−0.166590	−	0.986026i	$-0.553275\pi$
$739$	13.6045	−	6.55159i	0.500450	−	0.241004i	0.667040	+	0.745022i	$0.267561\pi$
$740$		0			0
$741$	0.840826	+	11.2200i	0.0308885	+	0.412178i
$742$		0			0
$743$	−9.34603	+	23.8133i	−0.342873	+	0.873626i	0.650586	+	0.759433i	$0.274523\pi$
$743$	−9.34603	+	23.8133i	−0.342873	+	0.873626i	−0.993459	+	0.114193i	$0.963572\pi$
$744$		0			0
$745$	12.0070	−	1.80976i	0.439903	−	0.0663046i
$746$		0			0
$747$	−54.0779	−	26.0425i	−1.97861	−	0.952847i
$748$		0			0
$749$	−9.32295	−	23.7545i	−0.340653	−	0.867970i
$750$		0			0
$751$	19.4680	−	18.0637i	0.710398	−	0.659153i	−0.239792	−	0.970824i	$-0.577079\pi$
$751$	19.4680	−	18.0637i	0.710398	−	0.659153i	0.950190	+	0.311671i	$0.100889\pi$
$752$		0			0
$753$	−36.1874	+	45.3776i	−1.31874	+	1.65365i
$754$		0			0
$755$	−11.3773	+	3.50943i	−0.414062	+	0.127721i
$756$		0			0
$757$	−35.7878	+	24.3997i	−1.30073	+	0.886824i	−0.997807	−	0.0661909i	$-0.978915\pi$
$757$	−35.7878	+	24.3997i	−1.30073	+	0.886824i	−0.302924	+	0.953015i	$0.597963\pi$
$758$		0			0
$759$	−12.0525	−	11.1831i	−0.437478	−	0.405920i
$760$		0			0
$761$	−2.59319	+	34.6036i	−0.0940029	+	1.25438i	0.728271	+	0.685289i	$0.240324\pi$
$761$	−2.59319	+	34.6036i	−0.0940029	+	1.25438i	−0.822274	+	0.569092i	$0.807295\pi$
$762$		0			0
$763$	−0.765713	−	3.35481i	−0.0277207	−	0.121452i
$764$		0			0
$765$	−4.85170	−	1.49655i	−0.175414	−	0.0541079i
$766$		0			0
$767$	−4.45393	+	7.71443i	−0.160822	+	0.278552i
$768$		0			0
$769$	−12.9224	−	8.81031i	−0.465992	−	0.317708i	0.307462	−	0.951560i	$-0.400520\pi$
$769$	−12.9224	−	8.81031i	−0.465992	−	0.317708i	−0.773454	+	0.633852i	$0.781473\pi$
$770$		0			0
$771$	64.4672	+	9.71687i	2.32173	+	0.349944i
$772$		0			0
$773$	36.9024			1.32729			0.663643	−	0.748049i	$-0.269009\pi$
$773$	36.9024			1.32729			0.663643	+	0.748049i	$0.269009\pi$
$774$		0			0
$775$	−2.64711			−0.0950871
$776$		0			0
$777$	−50.7073	−	7.64289i	−1.81911	−	0.274187i
$778$		0			0
$779$	3.82127	+	2.60530i	0.136911	+	0.0933444i
$780$		0			0
$781$	5.43359	−	9.41125i	0.194429	−	0.336761i
$782$		0			0
$783$	60.1537	+	18.5549i	2.14972	+	0.663100i
$784$		0			0
$785$	2.28414	+	10.0074i	0.0815243	+	0.357181i
$786$		0			0
$787$	2.89420	−	38.6204i	0.103167	−	1.37667i	−0.669950	−	0.742406i	$-0.733684\pi$
$787$	2.89420	−	38.6204i	0.103167	−	1.37667i	0.773117	−	0.634263i	$-0.218696\pi$
$788$		0			0
$789$	46.0737	+	42.7502i	1.64027	+	1.52195i
$790$		0			0
$791$	−21.7069	+	14.7995i	−0.771808	+	0.526210i
$792$		0			0
$793$	16.3006	−	5.02806i	0.578851	−	0.178552i
$794$		0			0
$795$	−6.51436	+	8.16875i	−0.231041	+	0.289716i
$796$		0			0
$797$	−28.3808	+	26.3335i	−1.00530	+	0.932782i	−0.997740	−	0.0671893i	$-0.978597\pi$
$797$	−28.3808	+	26.3335i	−1.00530	+	0.932782i	−0.00755980	+	0.999971i	$0.502406\pi$
$798$		0			0
$799$	−5.02036	−	12.7917i	−0.177608	−	0.452537i
$800$		0			0
$801$	31.5583	+	15.1977i	1.11506	+	0.536984i
$802$		0			0
$803$	−28.2994	+	4.26544i	−0.998663	+	0.150524i
$804$		0			0
$805$	−2.58769	+	6.59334i	−0.0912042	+	0.232385i
$806$		0			0
$807$	0.538819	+	7.19005i	0.0189673	+	0.253102i
$808$		0			0
$809$	23.3522	−	11.2458i	0.821018	−	0.395382i	0.0242795	−	0.999705i	$-0.492271\pi$
$809$	23.3522	−	11.2458i	0.821018	−	0.395382i	0.796739	+	0.604324i	$0.206557\pi$
$810$		0			0
$811$	5.83095	+	10.0995i	0.204752	+	0.354641i	0.950054	−	0.312086i	$-0.101028\pi$
$811$	5.83095	+	10.0995i	0.204752	+	0.354641i	−0.745302	+	0.666728i	$0.767694\pi$
$812$		0			0
$813$	−13.1562	+	57.6411i	−0.461408	+	2.02156i
$814$		0			0
$815$	8.11406	+	10.1747i	0.284223	+	0.356404i
$816$		0			0
$817$	−11.3598	−	8.63165i	−0.397431	−	0.301983i
$818$		0			0
$819$	−27.5099	−	34.4964i	−0.961274	−	1.20540i
$820$		0			0
$821$	1.54182	−	6.75515i	0.0538099	−	0.235756i	−0.940870	−	0.338767i	$-0.889990\pi$
$821$	1.54182	−	6.75515i	0.0538099	−	0.235756i	0.994680	+	0.103010i	$0.0328474\pi$
$822$		0			0
$823$	1.10308	+	1.91060i	0.0384511	+	0.0665993i	0.884611	−	0.466331i	$-0.154424\pi$
$823$	1.10308	+	1.91060i	0.0384511	+	0.0665993i	−0.846159	+	0.532930i	$0.821091\pi$
$824$		0			0
$825$	25.2719	−	12.1703i	0.879854	−	0.423715i
$826$		0			0
$827$	0.862574	+	11.5102i	0.0299946	+	0.400250i	0.991914	+	0.126915i	$0.0405077\pi$
$827$	0.862574	+	11.5102i	0.0299946	+	0.400250i	−0.961919	+	0.273335i	$0.911873\pi$
$828$		0			0
$829$	17.7959	−	45.3432i	0.618077	−	1.57483i	−0.187450	−	0.982274i	$-0.560022\pi$
$829$	17.7959	−	45.3432i	0.618077	−	1.57483i	0.805527	−	0.592559i	$-0.201882\pi$
$830$		0			0
$831$	44.0671	−	6.64204i	1.52867	−	0.230410i
$832$		0			0
$833$	−15.9157	−	7.66459i	−0.551446	−	0.265562i
$834$		0			0
$835$	1.84743	+	4.70716i	0.0639328	+	0.162898i
$836$		0			0
$837$	3.47246	−	3.22197i	0.120026	−	0.111368i
$838$		0			0
$839$	3.63297	−	4.55560i	0.125424	−	0.157277i	−0.715155	−	0.698966i	$-0.753644\pi$
$839$	3.63297	−	4.55560i	0.125424	−	0.157277i	0.840579	+	0.541690i	$0.182215\pi$
$840$		0			0
$841$	27.1085	−	8.36187i	0.934777	−	0.288340i
$842$		0			0
$843$	−80.3458	+	54.7789i	−2.76726	+	1.88668i
$844$		0			0
$845$	−4.35447	−	4.04036i	−0.149798	−	0.138993i
$846$		0			0
$847$	−2.23536	+	29.8288i	−0.0768078	+	1.02493i
$848$		0			0
$849$	−8.27329	−	36.2476i	−0.283938	−	1.24402i
$850$		0			0
$851$	−10.3059	−	3.17895i	−0.353282	−	0.108973i
$852$		0			0
$853$	−6.61073	+	11.4501i	−0.226347	+	0.392045i	−0.956723	−	0.291001i	$-0.906012\pi$
$853$	−6.61073	+	11.4501i	−0.226347	+	0.392045i	0.730376	+	0.683046i	$0.239345\pi$
$854$		0			0
$855$	−6.21902	−	4.24005i	−0.212686	−	0.145007i
$856$		0			0
$857$	−0.650028	−	0.0979759i	−0.0222045	−	0.00334679i	0.137931	−	0.990442i	$-0.455955\pi$
$857$	−0.650028	−	0.0979759i	−0.0222045	−	0.00334679i	−0.160135	+	0.987095i	$0.551193\pi$
$858$		0			0
$859$	−38.6154			−1.31754			−0.658771	−	0.752344i	$-0.728923\pi$
$859$	−38.6154			−1.31754			−0.658771	+	0.752344i	$0.728923\pi$
$860$		0			0
$861$	−27.5150			−0.937708
$862$		0			0
$863$	−16.3367	−	2.46236i	−0.556108	−	0.0838198i	−0.135026	−	0.990842i	$-0.543112\pi$
$863$	−16.3367	−	2.46236i	−0.556108	−	0.0838198i	−0.421083	+	0.907022i	$0.638350\pi$
$864$		0			0
$865$	5.46092	+	3.72319i	0.185677	+	0.126592i
$866$		0			0
$867$	−22.0221	+	38.1434i	−0.747909	+	1.29542i
$868$		0			0
$869$	7.10549	+	2.19175i	0.241037	+	0.0743501i
$870$		0			0
$871$	0.205628	+	0.900916i	0.00696745	+	0.0305264i
$872$		0			0
$873$	4.44603	−	59.3281i	0.150475	−	2.00795i
$874$		0			0
$875$	−18.3942	−	17.0673i	−0.621836	−	0.576979i
$876$		0			0
$877$	−38.9124	+	26.5300i	−1.31398	+	0.895855i	−0.998589	−	0.0531078i	$-0.983087\pi$
$877$	−38.9124	+	26.5300i	−1.31398	+	0.895855i	−0.315389	+	0.948963i	$0.602135\pi$
$878$		0			0
$879$	20.8289	−	6.42487i	0.702542	−	0.216705i
$880$		0			0
$881$	16.0354	−	20.1078i	0.540246	−	0.677447i	−0.434523	−	0.900661i	$-0.643083\pi$
$881$	16.0354	−	20.1078i	0.540246	−	0.677447i	0.974770	+	0.223213i	$0.0716546\pi$
$882$		0			0
$883$	−5.99913	+	5.56638i	−0.201887	+	0.187324i	−0.774637	−	0.632407i	$-0.782067\pi$
$883$	−5.99913	+	5.56638i	−0.201887	+	0.187324i	0.572750	+	0.819730i	$0.305877\pi$
$884$		0			0
$885$	−3.30290	−	8.41564i	−0.111026	−	0.282889i
$886$		0			0
$887$	25.8411	+	12.4444i	0.867658	+	0.417842i	0.814101	−	0.580723i	$-0.197230\pi$
$887$	25.8411	+	12.4444i	0.867658	+	0.417842i	0.0535564	+	0.998565i	$0.482944\pi$
$888$		0			0
$889$	−16.3132	+	2.45882i	−0.547128	+	0.0824662i
$890$		0			0
$891$	−5.39392	+	13.7435i	−0.180703	+	0.460424i
$892$		0			0
$893$	−1.52236	−	20.3145i	−0.0509439	−	0.679799i
$894$		0			0
$895$	8.28261	−	3.98870i	0.276857	−	0.133327i
$896$		0			0
$897$	−7.03918	−	12.1922i	−0.235031	−	0.407086i
$898$		0			0
$899$	0.960619	−	4.20875i	0.0320385	−	0.140370i
$900$		0			0
$901$	−5.40425	−	6.77671i	−0.180042	−	0.225765i
$902$		0			0
$903$	84.7678	+	4.35832i	2.82090	+	0.145036i
$904$		0			0
$905$	1.19274	+	1.49564i	0.0396479	+	0.0497169i
$906$		0			0
$907$	2.08244	−	9.12375i	0.0691461	−	0.302949i	−0.928516	−	0.371293i	$-0.878914\pi$
$907$	2.08244	−	9.12375i	0.0691461	−	0.302949i	0.997662	+	0.0683442i	$0.0217716\pi$
$908$		0			0
$909$	32.4306	+	56.1714i	1.07566	+	1.86309i
$910$		0			0
$911$	−15.5906	+	7.50806i	−0.516541	+	0.248753i	−0.673945	−	0.738781i	$-0.735402\pi$
$911$	−15.5906	+	7.50806i	−0.516541	+	0.248753i	0.157404	+	0.987534i	$0.449687\pi$
$912$		0			0
$913$	−1.57394	−	21.0028i	−0.0520899	−	0.695092i
$914$		0			0
$915$	−6.32501	+	16.1159i	−0.209098	+	0.532774i
$916$		0			0
$917$	−74.8963	+	11.2888i	−2.47329	+	0.372789i
$918$		0			0
$919$	−32.5165	−	15.6591i	−1.07262	−	0.516546i	−0.187669	−	0.982232i	$-0.560093\pi$
$919$	−32.5165	−	15.6591i	−1.07262	−	0.516546i	−0.884950	+	0.465686i	$0.845808\pi$
$920$		0			0
$921$	12.3342	+	31.4270i	0.406425	+	1.03555i
$922$		0			0
$923$	6.82121	−	6.32916i	0.224523	−	0.208327i
$924$		0			0
$925$	11.4720	−	14.3854i	0.377196	−	0.472988i
$926$		0			0
$927$	0.773544	−	0.238607i	0.0254065	−	0.00783687i
$928$		0			0
$929$	11.9724	−	8.16266i	0.392803	−	0.267808i	−0.350772	−	0.936461i	$-0.614081\pi$
$929$	11.9724	−	8.16266i	0.392803	−	0.267808i	0.743575	+	0.668652i	$0.233129\pi$
$930$		0			0
$931$	−19.1971	−	17.8123i	−0.629161	−	0.583776i
$932$		0			0
$933$	5.50765	−	73.4945i	0.180312	−	2.40610i
$934$		0			0
$935$	−0.396447	−	1.73695i	−0.0129652	−	0.0568042i
$936$		0			0
$937$	10.8335	+	3.34170i	0.353916	+	0.109169i	0.466615	−	0.884461i	$-0.345473\pi$
$937$	10.8335	+	3.34170i	0.353916	+	0.109169i	−0.112699	+	0.993629i	$0.535950\pi$
$938$		0			0
$939$	25.1146	−	43.4998i	0.819584	−	1.41956i
$940$		0			0
$941$	−28.7723	−	19.6166i	−0.937949	−	0.639483i	−0.00519143	−	0.999987i	$-0.501652\pi$
$941$	−28.7723	−	19.6166i	−0.937949	−	0.639483i	−0.932758	+	0.360504i	$0.882605\pi$
$942$		0			0
$943$	−5.72223	−	0.862488i	−0.186342	−	0.0280865i
$944$		0			0
$945$	21.6237			0.703419
$946$		0			0
$947$	−34.7749			−1.13003			−0.565016	−	0.825080i	$-0.691130\pi$
$947$	−34.7749			−1.13003			−0.565016	+	0.825080i	$0.691130\pi$
$948$		0			0
$949$	−24.2319	−	3.65237i	−0.786600	−	0.118561i
$950$		0			0
$951$	−57.9815	−	39.5311i	−1.88018	−	1.28188i
$952$		0			0
$953$	−3.94892	+	6.83974i	−0.127918	+	0.221561i	−0.922870	−	0.385112i	$-0.874163\pi$
$953$	−3.94892	+	6.83974i	−0.127918	+	0.221561i	0.794952	+	0.606673i	$0.207496\pi$
$954$		0			0
$955$	−3.03886	−	0.937364i	−0.0983352	−	0.0303324i
$956$		0			0
$957$	10.1790	+	44.5972i	0.329041	+	1.44162i
$958$		0			0
$959$	1.01427	−	13.5344i	0.0327523	−	0.437050i
$960$		0			0
$961$	22.4865	+	20.8644i	0.725370	+	0.673045i
$962$		0			0
$963$	−28.0352	+	19.1141i	−0.903422	+	0.615943i
$964$		0			0
$965$	3.83023	−	1.18147i	0.123299	−	0.0380328i
$966$		0			0
$967$	2.09656	−	2.62900i	0.0674208	−	0.0845429i	−0.746978	−	0.664849i	$-0.768496\pi$
$967$	2.09656	−	2.62900i	0.0674208	−	0.0845429i	0.814398	+	0.580306i	$0.197067\pi$
$968$		0			0
$969$	6.94435	−	6.44342i	0.223085	−	0.206992i
$970$		0			0
$971$	17.7430	+	45.2084i	0.569400	+	1.45081i	0.868065	+	0.496450i	$0.165363\pi$
$971$	17.7430	+	45.2084i	0.569400	+	1.45081i	−0.298665	+	0.954358i	$0.596541\pi$
$972$		0			0
$973$	20.5261	+	9.88485i	0.658037	+	0.316894i
$974$		0			0
$975$	23.7498	−	3.57970i	0.760602	−	0.114642i
$976$		0			0
$977$	0.376939	−	0.960425i	0.0120593	−	0.0307267i	−0.924717	−	0.380655i	$-0.875699\pi$
$977$	0.376939	−	0.960425i	0.0120593	−	0.0307267i	0.936777	+	0.349928i	$0.113794\pi$
$978$		0			0
$979$	0.918509	+	12.2567i	0.0293557	+	0.391724i
$980$		0			0
$981$	−4.12238	+	1.98523i	−0.131618	+	0.0633837i
$982$		0			0
$983$	−27.8010	−	48.1527i	−0.886713	−	1.53583i	−0.843738	−	0.536756i	$-0.819650\pi$
$983$	−27.8010	−	48.1527i	−0.886713	−	1.53583i	−0.0429752	−	0.999076i	$-0.513684\pi$
$984$		0			0
$985$	−0.645049	+	2.82614i	−0.0205530	+	0.0900484i
$986$		0			0
$987$	75.5647	+	94.7551i	2.40525	+	3.01609i
$988$		0			0
$989$	17.4924	+	3.56354i	0.556226	+	0.113314i
$990$		0			0
$991$	26.0171	+	32.6244i	0.826461	+	1.03635i	0.998684	+	0.0512909i	$0.0163336\pi$
$991$	26.0171	+	32.6244i	0.826461	+	1.03635i	−0.172223	+	0.985058i	$0.555095\pi$
$992$		0			0
$993$	−15.9612	+	69.9308i	−0.506515	+	2.21919i
$994$		0			0
$995$	6.75813	+	11.7054i	0.214247	+	0.371087i
$996$		0			0
$997$	−15.0791	+	7.26169i	−0.477558	+	0.229980i	−0.657148	−	0.753762i	$-0.728237\pi$
$997$	−15.0791	+	7.26169i	−0.477558	+	0.229980i	0.179589	+	0.983742i	$0.442523\pi$
$998$		0			0
$999$	2.46056	+	32.8339i	0.0778486	+	1.03882i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.bg.c.353.3		36
4.3	odd	2		43.2.g.a.9.2	✓	36
12.11	even	2		387.2.y.c.181.2		36
43.24	even	21	inner	688.2.bg.c.497.3		36
172.67	odd	42		43.2.g.a.24.2	yes	36
172.115	even	42		1849.2.a.o.1.14		18
172.143	odd	42		1849.2.a.n.1.5		18
516.239	even	42		387.2.y.c.325.2		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.g.a.9.2	✓	36	4.3	odd	2
43.2.g.a.24.2	yes	36	172.67	odd	42
387.2.y.c.181.2		36	12.11	even	2
387.2.y.c.325.2		36	516.239	even	42
688.2.bg.c.353.3		36	1.1	even	1	trivial
688.2.bg.c.497.3		36	43.24	even	21	inner
1849.2.a.n.1.5		18	172.143	odd	42
1849.2.a.o.1.14		18	172.115	even	42

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	688.bg (of order \(21\), degree \(12\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.93359 + 0.442167i) q^{3} +(-0.492700 - 0.335917i) q^{5} +(-2.18154 + 3.77854i) q^{7} +(5.54371 + 1.71001i) q^{9} +O(q^{10})\)	\(q+(2.93359 + 0.442167i) q^{3} +(-0.492700 - 0.335917i) q^{5} +(-2.18154 + 3.77854i) q^{7} +(5.54371 + 1.71001i) q^{9} +(0.452993 + 1.98469i) q^{11} +(-0.130264 + 1.73826i) q^{13} +(-1.29685 - 1.20330i) q^{15} +(1.21261 - 0.826746i) q^{17} +(2.07906 - 0.641305i) q^{19} +(-8.07048 + 10.1201i) q^{21} +(-1.99563 + 1.85167i) q^{23} +(-1.69679 - 4.32336i) q^{25} +(7.48807 + 3.60606i) q^{27} +(7.48963 - 1.12888i) q^{29} +(0.208229 - 0.530558i) q^{31} +(0.451329 + 6.02256i) q^{33} +(2.34412 - 1.12887i) q^{35} +(1.98083 + 3.43090i) q^{37} +(-1.15074 + 5.04173i) q^{39} +(1.32534 + 1.66193i) q^{41} +(-3.81986 - 5.32998i) q^{43} +(-2.15696 - 2.70475i) q^{45} +(2.08348 - 9.12834i) q^{47} +(-6.01823 - 10.4239i) q^{49} +(3.92287 - 1.88915i) q^{51} +(-0.441351 - 5.88942i) q^{53} +(0.443502 - 1.13002i) q^{55} +(6.38267 - 0.962032i) q^{57} +(4.60418 + 2.21726i) q^{59} +(-3.57526 - 9.10961i) q^{61} +(-18.5551 + 17.2167i) q^{63} +(0.648092 - 0.812681i) q^{65} +(0.506577 - 0.156258i) q^{67} +(-6.67310 + 4.54964i) q^{69} +(-3.91319 - 3.63091i) q^{71} +(-1.05058 + 14.0191i) q^{73} +(-3.06604 - 13.4332i) q^{75} +(-8.48745 - 2.61803i) q^{77} +(1.82633 - 3.16330i) q^{79} +(5.99228 + 4.08546i) q^{81} +(-10.2305 - 1.54199i) q^{83} -0.875173 q^{85} +22.4706 q^{87} +(5.97021 + 0.899865i) q^{89} +(-6.28390 - 4.28429i) q^{91} +(0.845453 - 1.46437i) q^{93} +(-1.23978 - 0.382420i) q^{95} +(-2.28198 - 9.99799i) q^{97} +(-0.882576 + 11.7772i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9}+O(q^{10}) \) 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9	\( 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9} + 4 q^{11} + 3 q^{15} - 10 q^{17} - 10 q^{19} - 21 q^{21} - 4 q^{23} - 2 q^{25} + 4 q^{27} + 9 q^{29} - 40 q^{31} - 11 q^{33} - 11 q^{35} - 19 q^{37} + q^{39} - 28 q^{41} + 8 q^{43} - 46 q^{45} + 30 q^{47} + 6 q^{49} - 57 q^{51} - 24 q^{53} - 14 q^{55} + 52 q^{57} + q^{59} - 14 q^{61} - 47 q^{63} + 38 q^{65} - 66 q^{67} - 7 q^{69} + 33 q^{71} + 29 q^{73} + 55 q^{75} - 27 q^{77} + 17 q^{79} + 38 q^{81} + 23 q^{83} - 56 q^{85} + 86 q^{87} - 19 q^{89} + 13 q^{91} - 30 q^{93} - q^{95} - 31 q^{97} + 31 q^{99}+O(q^{100}) \) 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9 + 4 * q^11 + 3 * q^15 - 10 * q^17 - 10 * q^19 - 21 * q^21 - 4 * q^23 - 2 * q^25 + 4 * q^27 + 9 * q^29 - 40 * q^31 - 11 * q^33 - 11 * q^35 - 19 * q^37 + q^39 - 28 * q^41 + 8 * q^43 - 46 * q^45 + 30 * q^47 + 6 * q^49 - 57 * q^51 - 24 * q^53 - 14 * q^55 + 52 * q^57 + q^59 - 14 * q^61 - 47 * q^63 + 38 * q^65 - 66 * q^67 - 7 * q^69 + 33 * q^71 + 29 * q^73 + 55 * q^75 - 27 * q^77 + 17 * q^79 + 38 * q^81 + 23 * q^83 - 56 * q^85 + 86 * q^87 - 19 * q^89 + 13 * q^91 - 30 * q^93 - q^95 - 31 * q^97 + 31 * q^99

Introduction

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